Real-time observation of non-equilibrium phonon-electron energy and angular momentum flow in laser-heated nickel

Identifying the microscopic nature of non-equilibrium energy transfer mechanisms among electronic, spin, and lattice degrees of freedom is central to understanding ultrafast phenomena such as manipulating magnetism on the femtosecond timescale. Here, we use time- and angle-resolved photoemission spectroscopy to go beyond the often-used ensemble-averaged view of non-equilibrium dynamics in terms of quasiparticle temperature evolutions. We show for ferromagnetic Ni that the non-equilibrium electron and spin dynamics display pronounced variations with electron momentum, whereas the magnetic exchange interaction remains isotropic. This highlights the influence of lattice-mediated scattering processes and opens a pathway toward unraveling the still elusive microscopic mechanism of spin-lattice angular momentum transfer.


Supplementary text
Time-dependent shift of the valence-band maximum (VBM) in tr-ARPES Figure S1 shows the momentum dependence of the maximum energy of transient occupied valence states, EVBM, defined as the energy where the photoemission intensity near EF is reduced by half, for selected delay times after optical excitation.In the ground-state (i.e., when the probe pulses arrive at the sample at a delay time, t, before the pump, i.e. t < 0), we find that the highest values of EVBM occur for momenta where real bands are near EF.This occurs when real bands cross the Fermi level, i.e., for sp-bands in the G-K direction near k = 1.0 Å -1 (Figs.S1D, H) or for 3d-bands in the G −  !! # " direction near k = 1.1 Å -1 and 1.6 Å -1 in Figs.S1B, F. High values of EVBM also occur near the W-point (Figs.S1A, E) when the majority 3d-band approaches EF without crossing it.300 fs after laser excitation, EVBM has increased at the W-point by almost 80 meV.For larger delay times EVBM slowly decreases by 30 meV at 1.8 ps.A monotonic increase of EVBM is observed at the sp-band Fermi crossing and for 3d-states near the  $!!point.Such a significant momentum dependence of electronic excitations has not been reported before for Ni (11,21,28) or other ferromagnets (46,47).The significantly different shift of EVBM observed for different k-points excludes space charge effects as a cause of this behavior.For other materials, an increase of EVBM has been attributed to a change of the global chemical potential, i.e., to increased band occupations (39,40).

Core-level measurements
To corroborate the observed chemical potential shifts in the valence band, we also measured time-resolved 3p core level photoemission spectra for the same fluence conditions.The unpumped spectrum in Fig. S7A is averaged between time delays -0.4 ps < t < 0.2 ps.It clearly shows spin-orbit-split peaks whose binding energy was determined by fitting two Voigt functions for each core-level.The background for the core-level spectra has been calculated using the statistics-sensitive non-linear iterative peak-clipping (SNIP) algorithm (43,44).We obtain values for binding energies of the spin-orbit-split 3p1/2 and 3p3/2 peaks of 68.0 and 66.2 eV with a width of 1.7 and 1.8 eV, respectively which agree well with the reports in the literature (45).
Upon pumping at a pump fluence of 3.7 mJ/cm 2 we observe a slight attenuation of the amplitude as shown in Fig. S12.This amplitude reduction could be due to enhanced scattering of photoemitted electrons with the increase of phonon population.The change in peak binding energy with pump-probe time delay is illustrated in Fig. S7B for a time-delay of 1.8 ps.In order to reduce the number of fit parameters we keep the ratio of peak amplitudes; peak width and spin-orbit splitting the same at all time delays.Figure S7B also shows the comparison of the measured difference spectra with two fit conditions which are 2% (1.7%) reduction in amplitude and 12 meV (15 meV) binding-energy shifts towards the Fermi level.The change in the amplitude and peak shift have different signatures in the difference photoemission spectra with absorptive and dispersive behavior respectively.As shown in Fig. S7B, 15 meV shift and 1.7% amplitude decrease results in better fit (gray plot) in the slope around 66 eV but introduces a positive feature around 65 eV whereas blue fits better at 65 eV at the cost of poor fit in the slope.

Correction of the Light Assisted Photoemission Effect (LAPE)
The LAPE is a transient process occurring only during the temporal overlap of intense infra-red (IR) pump and XUV probe (42).LAPE appears as a step in the photoemission intensity near Fermi level and is described as the laser (IR) dressing of the emitted photoelectrons in the continuum modifying the photoemission spectrum as shown in Fig S9 .Above the Fermi level additional photoemission intensity appears due to the absorption of an IR photon by an XUV emitted photoelectron.The above-Fermi intensity distribution roughly follows the direct photoemission spectrum at lower binding energy shifted by the laser photon energy of 1.2 eV.The change in intensity above Fermi level (red line in Fig. S9A) relative to the ground-state photoemission spectra (dashed black lines) is a sensitive measure of the temporal duration of pump and probe pulses (see Fig. S10).We obtain values of 227±2 fs for the convoluted pump and probe pulse duration.For analyzing the spectral lineshape of photoemission spectra in temporal pump-probe overlap it is necessary to subtract the LAPE induced lineshape changes.Since we only analyze the lineshape in the vicinity of the Fermi level it is possible to remove the LAPE modifications as indicated in Fig. S9B.The LAPE corrected spectrum shown in red in Fig. 9B is very similar to the unpumped spectrum justifying the correction procedure in the binding energy range of ±1 eV around the Fermi level where significant effects due to pump laser are expected to occur.This correction procedure is then repeated for other time delays where LAPE is present (i.e. for delay times between -0.2 ps < t < 0.2 ps) to extract the pump induced dynamical signal.

Fitting valence photoemission spectra
We used the fit functions described in Eqs.(S1-S3) for modelling the photoemission signal for k-dependent density of states.The fit function generally includes the product of density of states with Fermi-Dirac distribution convoluted with a gaussian function in Eq. (S1).The Fermi-Dirac distribution determines the filling of the density of states while the gaussian convolution mimics the instrumental resolution.The evolution of the Fermi function for different k-points provides information about the change in the chemical potential and electronic temperature induced broadening around the Fermi level.The measured photoemission intensity is then where  % =  −  ) , is the electron energy relative to the Fermi level,  ) . &'( ,  * , µ and  + represent the density of states, Fermi-Dirac distribution, chemical potential and electronic temperature respectively.⊗ represents the convolution of a gaussian function, G, with d as its FWHM describing the energy resolution of the experiment. To extract the dynamics of exchange splitting from k-points with minority and majority d-bands, the density of states for up and down spin d-bands is produced by using Lorentzian functions Eq. (S2) as with  $ ,  +0   representing the energy position, exchange splitting and width of Lorentzian functions for minority and majority d-bands.The electronic temperature for these k-points are the values extracted from the fitting of sp-bands.The delay-independent background for d-bands is extracted using the Shirley method (30).The extracted energy shift of majority band for different k-points has been evaluated using two exponential functions convolved with temporal resolution ( 3 ) shown in Eq. ( S3) as where  &+-15 and  6+-15 are demagnetization and remagnetization time constants, respectively, erf is the error function., where  -12,-./are the occupations on majority and minority d-bands determined from Fig. 3 and  $ represents the respective occupation numbers before laser excitation.

Fig. S1 .
Fig. S1.Changes in the photoemission signal from states near the Fermi level, EF. (A-D) Energy position of the top-most occupied valence-band electronic state, EVBM, relative to the Fermi level, EF, along the respective directions for various pump-probe time delays.EVBM was chosen as the inflection point of data such as the ones shown in Fig. 2. (E-H) Momentum resolved ARPES intensity (color shading) and calculated electronic state dispersions (lines) along L-W, G- !!" #, L-U and G-K Brillouin-zone directions (see Fig. 1C), respectively.Data were obtained for a pump fluence of 3.7 mJ/cm 2 .

Fig. S5 .
Fig. S5.ARPES from states near the Fermi level in Ni(111).(A) photoemission intensity from states near EF in the G-K plane of the Ni(111) Brillouin zone (see Fig. 1C) (B) same as in (A) but for the L-U-W plane in Fig. 1C.The shown regions bounded by dotted lines along different k-directions are used for the analysis in Fig. 2 and Fig S2.

Fig. S6 .
Fig. S6.Temporal evolution of the 3d exchange splitting at different k-points.Energy positions, DEmaj, of the majority spin-bands for tr-ARPES spectra at the W, U and  !!" # -points obtained by fits as shown in Fig. 4 of the main paper for pump fluences of (A) 2.2 mJ/cm 2 and (B) 3.7 mJ/cm 2 .Dashed black lines represent the fits to the data according to Eq. (S3) with demagnetization times, tdemag, of 205±50 fs and 200±70 fs, and remagnetization times, tremag, of 3.0±0.5 and 2.9±0.7 ps for 2.2 and 3.7 mJ/cm 2 , respectively.

Fig. S7 .
Fig. S7.Temporal evolution of Ni(111) 3p core-level photoemission.(A) 3p corelevel photoemission spectra measured with 204 eV shown in black markers.The dotted black line is the calculated background using the SNIP method described in the supplementary text.The dashed line is the fit obtained using two Voigt functions representing the spin-orbit-split 3p1/2 and 3p3/2 core levels (colored solid lines).(B) Difference between pumped and unpumped corelevel photoemission spectra at a pump fluence of 3.7 mJ/cm 2 (dash-dotted line and circles).Blue and gray lines are the fits obtained for two conditions as described in the supplementary text.(C) Temporal evolution of the core-levelshift (red line and squares) towards the Fermi level compared with chemical potential change for the indicated k-points.

Fig. S8 .
Fig. S8.Tr-ARPES of Ni(111)/W(110) of different Ni thicknesses.(A) Unpumped angle-integrated photoemission spectra taken for Ni films of 2 nm thickness (blue lines and open squares) and 9 nm thickness at 205 eV photon energy (red lines and solid circles).At 205 eV photon energy similar parts of the Ni BZ are probed as at 74 eV.(B) Pumped angle-integrated photoemission spectra taken for Ni films of 2 and 9 nm thickness as in (A) averaged over ±0.3 ps around a pump-probe time delay of 1 ps.(C) Measured electron temperatures, Te, vs. delay time for the 2 and 9 nm thick Ni films.The pump fluence of the 2 nm film was 3.7 mJ/cm 2 as in the main paper.The pump fluence of the 9 nm film was adjusted to achieve a similar maximum Te value, i.e. a similar deposited pump laser energy.(D) Chemical potential change for the two Ni thicknesses with pump-probe delay.

Fig. S9 .
Fig. S9.Correction of the Light Assisted Photoemission Effect (LAPE).A)The black dashed line is the unpumped photoemission spectrum and the red solid line is the pumped photoemission spectrum with pump and probe pulses in temporal overlap.The deviation of both spectra is due to the LAPE effect.(B)The pumped spectrum after LAPE correction (solid red line) is compared with black unpumped spectrum (dashed black line).LAPE correction is done by first subtracting an offset to remove the step above the Fermi level and then multiplying the remaining spectrum by a factor to match it to the unpumped spectrum.

Fig. S10 .
Fig. S10.Temporal instrumental resolution.The black markers represent the photoemission signal above the Fermi level in the LAPE region (see Fig. S9 A).The red dashed line is a gaussian fit with a FWHM of 227±2 fs.

Fig. S11 .
Fig. S11.Extracted fit parameters at W, U and   # points with exchange split dbands for 3.7 (red) and 2.2 (blue) mJ/cm 2 pump fluence.(A-C) chemical potential shift, (D-F) shift of majority band due to exchange collapse.(G-I) full width half maximum (FWHM) broadening of majority bands due to laser excitation as a function of pump probe delay.

Fig. S12 .
Fig. S12.Laser induced reduction of 3p peaks.This figure shows the relative change in the mean amplitude of the 3p peaks for different pump-probe delays.There is a reduction in the amplitude of 3p peaks by ~1.7-2% after laser excitation.

Fig. S13 .
Fig. S13.Calculated transient phonon occupations.This figure shows the differences of transient phonon occupation relative to those in thermal equilibrium 1 ps after fs laser excitation in the L-W-U (A-C) and G-K (D-F) planes of the Ni Brillouin zone (see Fig. 1B).Panels (A, B, D, E) show phonons with transverse polarization while (C, F) display longitudinal phonon polarization.